Calculus of median and bisector

Question

In the triangle ABC we draw the median segment AD and the bisector BE. We know that AB=7, BC=18 ed EA=ED. How is the length of AC?

Answer 1

Since $\measuredangle ABE=\measuredangle DBE$ and $AE=DE$,

we see that $BAED$ is cyclic, which says that $\Delta ABC\sim\Delta DEC$.

Let $AE=7x$.

Hence, $ED=7x$ and $EC=18x$ and since $$\frac{ED}{AB}=\frac{DC}{AC},$$ we obtain $$\frac{7x}{7}=\frac{9}{25x},$$ which gives $x=\frac{3}{5}$ and $AC=25\cdot\frac{3}{5}=15$.

Done!